Franziska and Safua play the game Halvsies. The game is set up with one pile of $5$ stones,

one pile of $8$ stones, and one pile of $11$ stones. Franziska, who's younger than Safua,

always goes first. On their turn, a player does one of the following:

They may remove stones from a single pile. They remove at least one stone and no

more than half of the pile $($rounded up$).$

They may split one pile, consisting of more than one stone, into two new piles as

evenly as possible.

Whichever player removes the last stone is the winner.

Of course, Halvsies could be played starting with any number of piles and any numbers

of stones. Answer the following questions to generate some avenues of exploration and

perhaps solve the original game.

$($a$)$ Show that if the game were set up with two piles of equal size, Safua can guarantee

a win. Conclude that if they played Halvsies with a single pile with an even number

of stones, Franziska can guarantee a win.

$($b$)$ Suppose Safua wins in a single$-$pile game with A stones and also in a single$-$pile

game with $B$ stones. Does she win in the game which starts with one pile of A and

one pile of $B?$

$($c$)$ Suppose Franziska wins in a single$-$pile game with A stones and also in a single$-$pile

game with $B$ stones, and $AB.$ Does she win in the game which starts with one

pile of A and one pile of $B?$

$($d$)$ Hopefully there are some questions that naturally arose that you can pursue now.

Can you determine who wins in other games of Halvsies? Who can guarantee a win

in the original game set$-$up$?$